Subgroups — C₂×#11750970 #14416479

Order 1920.

1216 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#91782	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#91093	C₂	1	2	yes	maximal, complemented
6	#11750970	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65158	—	3	1	no	maximal, Hall
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×C₄×D₁₅	—	2	2	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#6714	—	2	2	no	—
13	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₂×D₁₂₀	C₄	1	8	yes	complemented
16	C₂×D₁₂₀	C₂²	1	4	yes	—
17	#91782	—	2	4	no	—
18	#91782	C₂²	1	8	yes	complemented
19	C₃×C₅×#106	C₂²	1	4	yes	complemented
20	#91099	C₂²	1	4	yes	complemented
21	#91093	C₂²	1	8	yes	complemented
22	C₂×#4869	—	5	1	no	maximal, Hall
23	C₂²×D₄₀	—	3	1	no	—
24	C₂×#4039	—	3	2	no	—
25	C₂×C₅×#106	D₃	1	1	yes	characteristic, complemented
26	C₂×#3947	—	3	1	no	—
27	C₂×#3941	—	3	2	no	—
28	#65158	—	3	8	no	—
29	C₂³×D₁₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₅	D₄	1	2	yes	—
33	C₂×D₆₀	—	2	2	no	—
34	C₂×D₆₀	—	2	4	no	—
35	C₂×D₆₀	C₂×C₄	1	8	yes	complemented
36	C₂×D₆₀	C₂³	1	4	yes	—
37	C₂×C₄×D₁₅	—	2	8	no	—
38	C₂×C₄×D₁₅	—	4	8	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	#6714	—	2	8	no	—
44	C₄×Dic₁₅	D₄	1	2	yes	—
45	C₄×Dic₁₅	C₂³	1	2	yes	—
46	C₂×C₃⋊#168	—	2	2	no	—
47	C₂×C₃⋊#168	C₂³	1	2	yes	—
48	D₁₂₀	C₂×C₄	1	16	yes	complemented
49	C₂²×D₂₄	—	5	1	no	—
50	C₂×#893	—	5	2	no	—
51	C₂×C₃×#106	D₅	1	1	yes	characteristic, complemented
52	C₂×#815	—	5	1	no	—
53	C₂×#809	—	5	2	no	—
54	#4869	—	5	8	no	—
55	C₂²×D₂₀	—	3	2	no	—
56	C₂²×C₄×D₅	—	6	2	no	—
57	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
59	C₂×#685	—	6	2	no	—
60	C₂×C₄×Dic₅	—	3	1	no	—
61	C₂²×#168	—	3	1	no	—
62	C₂×D₄₀	—	3	8	no	—
63	C₂×D₄₀	—	3	4	no	—
64	#4039	—	3	8	no	—
65	#4039	—	6	4	no	—
66	C₅×#106	C₂×D₃	1	4	yes	complemented
67	#3947	—	3	4	no	—
68	#3941	—	3	8	no	—
69	C₂×#442	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₁₅	—	2	8	no	—
72	C₂²×D₁₅	—	2	4	no	—
73	C₂²×D₁₅	—	4	4	no	—
74	C₂²×D₁₅	—	4	4	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	4	4	no	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	4	no	—
82	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
83	D₆₀	—	2	8	no	—
84	D₆₀	—	2	4	no	—
85	D₆₀	C₂²×C₄	1	8	yes	complemented
86	C₄×D₁₅	—	4	16	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
88	C₃⋊#168	—	2	2	no	—
89	C₂²×D₁₂	—	5	2	no	—
90	C₂²×C₄×D₃	—	10	2	no	—
91	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
92	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
93	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
94	C₂×C₄×Dic₃	—	5	1	no	—
95	C₂²×C₃⋊C₈	—	5	1	no	—
96	C₂×D₂₄	—	5	8	no	—
97	C₂×D₂₄	—	5	4	no	—
98	#893	—	5	8	no	—
99	#893	—	10	4	no	—
100	C₃×#106	C₂×D₅	1	4	yes	complemented
101	#815	—	5	4	no	—
102	#809	—	5	8	no	—
103	C₂³×D₅	—	6	2	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	3	2	no	—
107	C₂×D₂₀	—	3	8	no	—
108	C₂×D₂₀	—	3	4	no	—
109	C₂×D₂₀	—	6	2	no	—
110	C₂×D₂₀	—	6	4	no	—
111	C₂×C₄×D₅	—	6	8	no	—
112	C₂×C₄×D₅	—	12	8	no	—
113	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
114	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
117	#685	—	6	8	no	—
118	C₄×Dic₅	—	3	2	no	—
119	C₄×Dic₅	—	3	2	no	—
120	C₂×#168	—	3	2	no	—
121	C₂×#168	—	6	2	no	—
122	D₄₀	—	3	16	no	—
123	C₂²×D₈	—	15	1	no	—
124	C₂×C₄×D₄	—	15	2	no	—
125	#442	—	15	8	no	—
126	C₂×#106	D₁₅	1	1	yes	characteristic, complemented
127	C₂×#102	—	15	2	no	—
128	C₂×#97	—	15	1	no	—
129	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
130	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₅	—	2	8	no	—
133	C₂×D₁₅	—	4	4	no	—
134	C₂×D₁₅	—	4	4	no	—
135	C₂×D₁₅	—	4	16	no	—
136	C₃×C₄×C₅	—	4	4	no	—
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
140	Dic₁₅	—	2	4	no	—
141	C₂³×D₃	—	10	2	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×Dic₃	—	5	2	no	—
145	C₂×D₁₂	—	5	8	no	—
146	C₂×D₁₂	—	5	4	no	—
147	C₂×D₁₂	—	10	2	no	—
148	C₂×D₁₂	—	10	4	no	—
149	C₂×C₄×D₃	—	10	8	no	—
150	C₂×C₄×D₃	—	20	8	no	—
151	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
152	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
155	(C₂×D₃)⋊C₄	—	10	8	no	—
156	C₄×Dic₃	—	5	2	no	—
157	C₄×Dic₃	—	5	2	no	—
158	C₂×C₃⋊C₈	—	5	2	no	—
159	C₂×C₃⋊C₈	—	10	2	no	—
160	D₂₄	—	5	16	no	—
161	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
162	C₂²×D₅	—	6	8	no	—
163	C₂²×D₅	—	6	4	no	—
164	C₂²×D₅	—	12	4	no	—
165	C₂²×D₅	—	12	4	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	4	4	no	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#239	1	2	yes	—
171	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
172	C₂×Dic₅	—	3	4	no	—
173	C₂×Dic₅	—	6	4	no	—
174	D₂₀	—	3	8	no	—
175	D₂₀	—	6	8	no	—
176	D₂₀	—	6	4	no	—
177	C₄×D₅	—	12	16	no	—
178	C₅×C₈	C₂×C₄×D₃	1	4	yes	complemented
179	#168	—	6	2	no	—
180	C₂²×D₄	—	15	2	no	—
181	C₂³×C₄	—	30	2	no	—
182	C₂×D₈	—	15	8	no	—
183	C₂×D₈	—	15	4	no	—
184	C₂²×C₈	—	15	1	no	—
185	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
186	C₄×D₄	—	15	8	no	—
187	C₄×D₄	—	30	4	no	—
188	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
189	C₂×C₂²⋊C₄	—	30	2	no	—
190	C₂×C₄²	—	15	1	no	—
191	#106	C₂×D₁₅	1	4	yes	complemented
192	#102	—	15	8	no	—
193	#97	—	15	4	no	—
194	C₂×C₃×C₅	#442	1	4	yes	complemented
195	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
197	D₁₅	—	4	8	no	—
198	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
199	C₂²×D₃	—	10	4	no	—
200	C₂²×D₃	—	10	8	no	—
201	C₂²×D₃	—	20	4	no	—
202	C₂²×D₃	—	20	4	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	4	4	no	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
206	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	#713	1	2	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₂×Dic₃	—	5	4	no	—
210	C₂×Dic₃	—	10	4	no	—
211	D₁₂	—	5	8	no	—
212	D₁₂	—	10	4	no	—
213	D₁₂	—	10	8	no	—
214	C₄×D₃	—	20	16	no	—
215	C₃×C₈	C₂×C₄×D₅	1	4	yes	complemented
216	C₃⋊C₈	—	10	2	no	—
217	C₂²×C₅	#893	1	2	yes	—
218	C₂²×C₅	#914	1	4	yes	—
219	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
220	C₂×D₅	—	6	8	no	—
221	C₂×D₅	—	12	16	no	—
222	C₂×D₅	—	12	4	no	—
223	C₂×D₅	—	12	4	no	—
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	#893	1	2	yes	—
226	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄×C₅	C₂×#239	1	1	yes	characteristic
228	Dic₅	—	6	4	no	—
229	C₂⁴	—	30	2	no	—
230	C₂×D₄	—	15	8	no	—
231	C₂×D₄	—	15	4	no	—
232	C₂×D₄	—	30	2	no	—
233	C₂×D₄	—	30	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	15	2	no	—
236	C₂²×C₄	—	30	8	no	—
237	C₂²×C₄	—	60	8	no	—
238	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
239	D₈	—	15	16	no	—
240	C₂×C₈	—	15	2	no	—
241	C₂×C₈	—	30	2	no	—
242	C₂×C₈	C₄×D₁₅	1	4	yes	complemented
243	C₂×C₈	C₂²×D₁₅	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
246	C₂²⋊C₄	—	30	8	no	—
247	C₄²	—	15	2	no	—
248	C₄²	—	15	2	no	—
249	C₃×C₅	C₂×#442	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	#4039	1	2	yes	—
251	C₂²×C₃	#4060	1	4	yes	—
252	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
253	C₂×D₃	—	10	8	no	—
254	C₂×D₃	—	20	4	no	—
255	C₂×D₃	—	20	4	no	—
256	C₂×D₃	—	20	16	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	#4039	1	2	yes	—
259	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
260	C₃×C₄	C₂×#713	1	1	yes	characteristic
261	Dic₃	—	10	4	no	—
262	C₂×C₅	#4869	1	4	yes	complemented
263	C₂×C₅	C₂×#893	1	1	yes	characteristic
264	C₂×C₅	C₂×#914	1	2	yes	—
265	D₅	—	12	8	no	—
266	C₂³	—	30	8	no	—
267	C₂³	—	30	4	no	—
268	C₂³	—	60	4	no	—
269	C₂³	—	60	4	no	—
270	C₂³	D₄×D₁₅	1	1	yes	characteristic
271	D₄	—	15	8	no	—
272	D₄	—	30	8	no	—
273	D₄	—	30	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	15	4	no	—
277	C₂×C₄	—	30	4	no	—
278	C₂×C₄	—	60	16	no	—
279	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
280	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
281	C₂×C₄	#6821	1	2	yes	—
282	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
283	C₈	—	30	2	no	—
284	C₈	C₂×C₄×D₁₅	1	4	yes	complemented
285	C₂×C₃	#65158	1	4	yes	complemented
286	C₂×C₃	C₂×#4039	1	1	yes	characteristic
287	C₂×C₃	C₂×#4060	1	2	yes	—
288	D₃	—	20	8	no	—
289	C₅	C₂×#4869	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	30	8	no	—
291	C₂²	—	60	16	no	—
292	C₂²	—	60	4	no	—
293	C₂²	—	60	4	no	—
294	C₂²	#91782	1	2	yes	—
295	C₂²	#91803	1	4	yes	—
296	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	30	4	no	—
299	C₄	#91782	1	2	yes	—
300	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
301	C₄	C₂×#6821	1	1	yes	characteristic
302	C₃	C₂×#65158	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	60	8	no	—
304	C₂	#11750970	1	4	yes	direct product factor
305	C₂	C₂×#91782	1	1	yes	characteristic
306	C₂	C₂×#91803	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×#106	D₁₅
hypocenter	C₃×C₅	C₂×#442
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#91782	C₂	semidirect
3. C₂×C₃×C₅×#106	C₂	semidirect
4. C₂×#91099	C₂	semidirect
5. C₂×#91093	C₂	semidirect
6. #11750970	C₂	direct (N × H)
15. C₂×D₁₂₀	C₄	semidirect
18. #91782	C₂²	semidirect
19. C₃×C₅×#106	C₂²	semidirect
20. #91099	C₂²	semidirect
21. #91093	C₂²	semidirect
25. C₂×C₅×#106	D₃	semidirect
35. C₂×D₆₀	C₂×C₄	semidirect
39. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
48. D₁₂₀	C₂×C₄	semidirect
51. C₂×C₃×#106	D₅	semidirect
66. C₅×#106	C₂×D₃	semidirect
85. D₆₀	C₂²×C₄	semidirect
87. C₃×C₅×C₈	C₂²×C₄	semidirect
100. C₃×#106	C₂×D₅	semidirect
113. C₂×C₅×C₈	C₄×D₃	semidirect
126. C₂×#106	D₁₅	semidirect
151. C₂×C₃×C₈	C₄×D₅	semidirect
178. C₅×C₈	C₂×C₄×D₃	semidirect
191. #106	C₂×D₁₅	semidirect
194. C₂×C₃×C₅	#442	semidirect
215. C₃×C₈	C₂×C₄×D₅	semidirect
242. C₂×C₈	C₄×D₁₅	semidirect
249. C₃×C₅	C₂×#442	semidirect
262. C₂×C₅	#4869	semidirect
284. C₈	C₂×C₄×D₁₅	semidirect
285. C₂×C₃	#65158	semidirect
289. C₅	C₂×#4869	semidirect
302. C₃	C₂×#65158	semidirect
304. C₂	#11750970	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₁₂₀	2
C₂×#91782	2
C₂×C₃×C₅×#106	2
C₂×#91099	2
C₂×#91093	2
#11750970	2
C₂×#65158	3
C₂×#4869	5

Subgroups — C2×#11750970 #14416479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11750970 #14416479